All of the 3rd grade teachers and students from Springer went on a field trip to an art museum. Tickets were $$6.50$ each for teachers and $$3.50$ each for students, and the group paid $$48.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$13.00$ each for teachers and $$13.00$ each for students, and the group paid $$156.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+3.5y = 48}$ ${13x+13y = 156}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-13x-7y = -96}$ ${13x+13y = 156}$ Add the top and bottom equations together. $ 6y = 60 $ $ y = \dfrac{60}{6}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {6.5x+3.5y = 48}$ to find $x$ ${6.5x + 3.5}{(10)}{= 48}$ $6.5x+35 = 48$ $6.5x = 13$ $x = \dfrac{13}{6.5}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {13x+13y = 156}$ and get the same answer for $x$ ${13x + 13}{(10)}{= 156}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.